Using mathematical models to evaluate germination rate and seedlings length of chickpea seed (Cicer arietinum L.) to osmotic stress at cardinal temperatures

Cicer arietinum is the 3 most important cool season legume crop growing in vast arid and semi-arid regions of the world. A lab experiment was designed using hydrothermal time model (HTT) to investigate the chickpea seed germination (SG) behavior, cardinal temperatures and germination responses across fluctuating temperatures (Ts) and water potentials (Ψs). Seeds of chickpea var. NIFA 1995 were germinated at six constant Ts (7, 14, 21, 28, 35 and 42 ̊C) each having the following five water potentials: 0, -0.2, -0.4–0.6 and -0.8 MPa. Germination percentage (G%) decreased significantly at (*P� 0.05) from 86.7% at 28 ̊C in -0.2 MPa to 10% in -0.2 MPa at 7 ̊C. The germination rate (GR = 1/t50) against different T percentiles exhibited that linear increase was observed in the GR pattern above and below the To. Based on the confidence intervals of the model coefficients and (R : 0.96), the average cardinal temperatures were 4.7, 23 and 44.2 ̊C for the base (Tb), optimal (To) and ceiling (Tc) temperatures respectively. θT1 value was observed maximum at 28 ̊C in -0.2 MPa and decreases with decreasing Ψ (-0.8 MPa). In comparison with control, the θT2 value was also highest in -0.2 MPa at 28 ̊C. The thermal time (TT) concept is well fitted to germination fraction data in distilled water with an R value increasing 0.972. The hydro time constant (θH) increased with an increase in T to To and then decreased when T>To. The ѱb(50) irregularly varied with increasing T, σΨb was also recorded lowest (0.166 MPa) at 28 ̊C and highest (0.457 MPa) at 7 ̊C. Based on the statistical analysis, cardinal temperatures, hydrothermal time constant (θHTT) and germination findings the HTT gives an insight into the interactive effect of T and Ψ on seed germination time courses under varying environmental conditions. PLOS ONE PLOS ONE | https://doi.org/10.1371/journal.pone.0260990 December 17, 2021 1 / 16 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111


Introduction
Chickpea (Cicer arietinum L.) is the 3 rd most authoritative cool season grain legume crop after the Phaseolus vulgaris and Pisum sativum, which are grown in vast arid and semi-arid regions of the world [1]. It is documented as one of the famous and oldest pulses cultivated from ancient times both in Europe and in Asia. Chickpea seeds are a decisive and inexpensive source of highly digestible nutrients such as vitamins, proteins, carbohydrates, minerals and potentially health beneficial phytochemicals especially for people in underprivileged countries [2]. After Turkey and India, Pakistan is the third largest commercial producer of chickpeas with a 7% segment in global production [3,4]. In Pakistan, chickpea covers 73% of the land and shares 76% of the gross production [5,6]. SG is the first and most complex morphological and physiological phase in the succession of plant development where many factors such as temperature, water potential, drought and flood affect it [7]. Temperature (T) and water potential (C) are the two most important external features that affect seed germination rate and germination characteristics [8]. Depending on the water potential of the permeation medium, if it is reduced germination may be also stopped or decelerated [9]. GR is correlated with the differential magnitude between environment water potential and physiological threshold water potential for the emergence of seed radicle [10]. Similarly, when water is not limited, the soil temperature is the most important environmental factor affecting seed germination and consequently plant establishment in the field [11].
The earth's climate is changing drastically in recent decades and these climate changes affect all the physiological phases of plant growth and development [12]. This changing climate increases the intensity and frequency of the abiotic factors that contrarily reduced agricultural production [13]. In these abiotic factors, temperature is the key factor affecting seed germination parameters in all plants [14]. However, it is significant that each taxon requires a certain range of Ts for seed germination called cardinal temperatures [15]. These are the minimum or base temperature (T b ), maximum or ceiling temperature (T c ) at which germination rate is minimum and optimum temperature (T o ) at which maximum germination occurs in a short period [16].
Gummerson developed mathematical thermal time (TT), hydro time (HT) and hydrothermal time model (HTT) to study the eco-physiological responses of seed germination to T, C and their interaction (T × C) [17]. Models can be quite useful in forecasting seed emergence time, dormancy and its use in crop management. Seed germination in the TT model is summarized in three cardinal temperatures (base, ceiling and optimum temperature). Similarly, the HT model is also described in three parameters (a) base water potential (C b ) (b) hydro time constant (θH) and (c) Standard deviation of the base water potential (σC b ) [10]. The base water potential is the theoretical threshold describing the prevention of seed germination fraction. TT, HT and HTT are the best models for describing seed germination patterns with T, C and their interactions (T × C). In recent times, the hydrothermal time model has been extensively used by many researchers to individually and simultaneously inspect the results of both T and C on seed germination outlines in several plants such as safflower (Carthamus tinctorius L.) [18], carrot (Daucus carota) [19], watermelon (Citrullus vulgaris) [20] as well as in weedy species.
Therefore, the objective of the current research work was: (i) to inspect the SG of C. arietinum at several T s , C s and their interactions (T × C) (ii) to explain chickpea seed SG behavior using models of the thermal time, hydro time and hydrothermal time (iii) to determine SG cardinal T s and drought tolerance threshold (base water potential). To the best of our knowledge, this is the first research study from Pakistan related to the application of the HTT model to predict the germination patterns of C. arietinum under various T s and C s .

Seeds description and experiment protocol
Chickpea (Cicer arietinum L.) var. NIFA 1995 seeds were brought from Nuclear Institute for Food and Agriculture (NIFA), Peshawar, Khyber Pakhtunkhwa, Pakistan with a 95% viability rate. The variety is described as light brown, angular shaped with an average weight (0.22 ±2gm) and length (0.4±1cm). Before the experiment, the seeds were stored in sealed plastic bags at 4˚C in dark.
The seeds were surface sterilized with 95% ethanol solution for 3 minutes and then washed with distilled water and shade dried at room temperature. The experiment was conducted according to randomized complete block design (RCBD), at various C s and T s in an incubator (Memmert Beschickung-Loading-Model 100-800, Germany) from October to December 2020 at Plant Physiology Lab. Department of Botany, University of Peshawar [5,21]. The experiment comprised of a wide range of constant temperatures (7,14,21,28,35 and 42˚C) each having the following four Cs (distilled water), -0.2, -0.4-0.6 and -0.8 MPa. The polyethylene glycol (PEG6000; Merck, Germany) was used for the preparation of C s solutions as suggested by [22]. Five accelerated aging periods (APP) of 24, 48, 72, 96 and 120 hours were used.
The standard germination test was performed by placing 35 seeds on Wathman No. 1 filter paper in Petri dishes moistened with 5 ml distilled water and PEG6000 solutions. The Petri dishes were randomly placed in an incubator in dark except for reading times. Each treatment has three replicates. Seeds were considered to have germinated when the radicle was 1 mm in length and was observed each day. The experiment was completed after 120 hours and germinated seeds were removed and counted for their germination parameters.

Data analysis
The germination data was analyzed through repeated probit regression analysis defined by [23,24] which is based on TT, HT and HTT models. The germination rate (GR) was calculated as the inverse of the germination time for each percentile at each T or C.

Thermal time (TT)
The germination time course data of constant T s at each C was quantified through the thermal time (TT) concept.
At sub-optimal Ts the model was: and at supra-optimal Ts the model was: Hence the germination rate is the reverse of the seed emergence time, the Eqs (1) and (2) can be written as Eq (3): Where θT1 and θT2 are the thermal time constant, T is the actual temperature of germination, T b is the base temperature for germination fraction, g is the actual time to germination fraction, GR (g) is the germination rate of the seed population.

Hydro time (HT)
Modeling of SG in response to C s and accelerated aging period was analyzed using the hydro time model as suggested by [17]. θH determines the relationships between solute potential and germination rate in the same way as the thermal time model. The model can be expressed as: Where θH is the hydro time constant (MPa.h), C is actual water potential, C b(g) is the base water potential of germination fraction g (%), tg is the time of seed population for radicle emergence, GR (g) is the germination rate is the actual time to germination fraction g. These parameters can be used as an indicator to evaluate seed vigor and quality in a seed lot.

Hydrothermal time model (HTT)
The TT and HT models can be combined into a joint hydrothermal time model to predict and describe the SG patterns in response to varied T and C in a single seedbed. In terms of the HTT model, the germination time course at all T s and C in the sub-optimal T (from T b to T o ) can be described as: Where θHTT is the hydrothermal time constant (MPa h) that has a constant value of seed population. T o is the predicted optimum temperature for germination fraction. k T is Boltzmann constant (slope of C b(g) versus T when T>T o ).

Germination attributes
The following germination indices were calculated from the per day and cumulative germination, physical observation, radicle and plumule lengths and fresh and dry weight of the seedlings.
2.6.1. Germination percentage (G%). G% represent the total number of seeds germinated out of the total seeds sown in each Petri dish. This germination parameter was calculated from the formula used by Onofri et al. [24].

Germination Percentage G%
ð Þ ¼ Final number of seedling emerged Total number of seeds sown � 100 Eqð8Þ 2.6.2. Germination energy (GE). Seed germination energy was calculated by using the formula suggested by Thriunavukkarasu and his colleagues [25].

Mean germination time (MGT).
The MGT index showed that how fast the seeds emerged in a population. Small MGT value means seed population has a high rate and vice versa. This was calculated by the formula of Onofri et al. [24].

Coefficient of velocity of germination (CVG).
The CVG represents the velocity of germination of seeds in an experiment. This will increase with an upsurge in the frequency of germinated seeds. The highest theoretical CVG value will be obtained when all sown seeds germinated on the first day. This was calculated from the formula suggested by Shah et al. [12].
Coefficient of Velocity of Germination ðCVGÞ 2.6.5. Germination index (GI). The Germination index tells us about the germination percentage and speed of germination. GI was calculated from the methodology of Labrie et al. [26].
2.6.6. Germination rate index (GRI). The GRI represent the percent germination on respective day and time. It was calculated by using the method of Rashidi [27].
2.6.7. Seed vigor index-1 (SVI-1). The length of three seedlings from each pot was measured in cm and then calculated by using the formula suggested by Chen et al. [28].

Seed vigor index-II (SVI-II).
The dry weight of three seedlings from each pot was determined through electrical balance and then multiply with seed germination percentage as suggested by Chen et al. [28].
Seed Vigor IndexÀ II ðSVIÀ IIÞ ¼ Seedling dry weight � Germintion %age Eqð15Þ 2.6.9. Time to 50% germination (T50%). This index was developed to find out the time required for 50% seed germination. T50% was calculated through a mathematical formula developed by Hao et al. [29].

Timson germination index (TGI).
The TGI index represents the average number of seeds germinated per day. This was measured according to the method of Hao et al. [29].

Data analysis
The investigation of T s (thermal time), C s (hydro time) and their interactions (hydrothermal time model) on seed germination rate and germination attributes were analyzed through linear regression using IBM SPSS Statistics 26 and SigmaPlot Version 11.0. The basic statistical calculation was performed in excel software (S1 File). The values of the following parameters: ѱ b(50), σC b ; R 2 , SE, Sig, F and T-test were determined using linear probit regression analysis in SPSS. ORIGIN 2021 PC Corporation was used for plotting various graphs of germination fraction vs accelerated aging period and germination parameters against T and C.

Germination responses to fluctuating temperatures (TT model)
Initially, the increase in temperature amplitude endorsed germination rate and percentage of seeds but decreased as T exceeds certain limit. Table 1 showed that temperature had significant ( � P�0.05) effects on germination rate (GR = 1/t 50 ) and percentage of Cicer arietinum var NIFA-1995. Under optimum moisture (0 MPa) the highest G% was recorded at 28˚C and lowest at 7 and 42˚C (Fig 1A and 1F). In general, the minimum germination (10 and  comparison with control (Fig 1A and 1F). It means germination increased from 40% to 93.3% with increasing temperature up to 28˚C, then decreased again to 40% as the temperature exceeds from 28˚C (optimum T) to 42˚C. In general germination percentage increased with AAP and significantly ( � P�0.05) decreased with high T (Fig 1A-1F). According to the results a substantially high θT1 value was observed at 28˚C in -0.2 MPa and decreases with decreasing C (-0.8 MPa) irrespective of species investigated (Table 1). In comparison with control, θT2 value was also documented highest in (-0.2 MPa) at 28˚C. The TT concept is well fitted to germination fraction data in distilled water with R 2 value increasing 0.972 (Table 2). Plotting the germination rate against different T percentiles exhibited that linear increase was observed in GR (g) pattern above and below the T o (Table 1) . Using the same pattern but in opposite direction, the GR (g) values showed significant ( �� P�0.01) increment with decreasing C (more negative) at all Ts. The base water potential at 50 percentile (ѱ b(50) ) showed an irregular pattern with increasing temperature. Similarly, the F and T-test also showed a lop-sided behavior having no linear pattern, except that both values were recorded maximum at 28˚C (Table 2). Initially, these GR (g) responses are applied to compute the base, optimum and ceiling temperature values.

Germination responses to water potentials (ѱ s ) (HT model)
The variation in seed germination time courses against water potential was scrutinized separately at each T (R 2 ranges 0.661-0.972) using the hydro time model. θH, C, ѱ b and GR (g) parameters for our experimental data at each constant T and various ѱ were analyzed by this HT concept. The minimum G% of 10% was recorded at 7˚C in -0.8 MPa and the maximum 86.3% in -0.8 MPa at 28˚C (Fig 1A and 1D). Generally, the highest G% was observed in -0.2 MPa and lowest in -0.8 at all T (Fig 1A-1F). It is observed that decreasing osmotic potential of the imbibition solution significantly reduced and delayed germination percentage and this inhibition effect was more severe at a temperature above and below optimum T. Both of these values showed minute increment with accelerated aging periods. The highest hydro time constant (θH) (41.6 MPa h) value was recorded in -0.2 MPa at 28˚C and lowest at 7˚C in -0.8 MPa in comparison with control ( Table 1). The data set of θH resulted from water potentials and germination fractions of seed population were plotted against constant T. The median base water potential (ѱ b(50) ) values were irregularly varied with increasing temperature having values ranging from -1.28 to -0.58 MPa ( Table 2). The standard deviation (σC b ) of the base water potential at 50 percentiles was also recorded lowest (0.166 MPa) at 28˚C and highest (0.457 MPa) at 7˚C. The GR (g) of the HT concept showed a remarkable high value at 7˚C in -0.8 MPa (Table 2).

Germination responses to temperature and water potential (HTT model)
The seed germination responses to integrating the effect of temperature and water potential above the thermal threshold (T b ) and the hydro threshold (ѱ b ) can be quantified using the hydrothermal time model. The HTT concept predictability was maximum (R 2 = 0.960 at 7˚C) at suboptimal temperatures (T<T o ) than at supra optimal temperature (R 2 = 0.872) ( Table 2). The interaction of T and C pointedly affected G% and GR ( � P<0.05). The comparative results of the HTT model showed that the effect of water potential on seed germination is slightly high as compared to temperature. The highest hydrothermal time constant (θHTT) value was observed in -0.2 MPa at 28˚C as compare to control (0 MPa). A linear pattern of increasing θHTT was followed as T increases up to optimal and decreases as we move towards ceiling T. Across various water potentials, the θHTT values decrease with decreasing osmotic potential (negative ѱ). The estimated cardinal temperatures of the HTT concept were 4.7˚C for T b , T o is 23˚C and T c is 44.2˚C respectively in control ( Table 3). Data of the germination attributes were presented in 3D graphs plotted against T and ѱ. Statistical analysis and graphical presentation revealed a significant change at ( � P�0.05) in radicle length, plumule length, G%, GE, MGT, CVG, GI, GRI, SVI-I and II, TGI and T50% in comparison with control treatment (Figs 2-4). The parameters which were affected negatively by increasing water potential (more negative) and temperature includes radicle length, plumule length, G%, GE, CVG, GI, GRI, SVI-I, SVI-II and TGI (Figs 2A-2D, 3B-3D and 4A-4C). These indices were reported highest in -0.2 MPa at 28˚C. On the contrary, the parameters MGT and T50% were reported maximum in -0.8 MPa at 42˚C (Figs 3A and 4D). The increasing pattern in these two indices is opposite with osmotic potential and temperature in comparison with above.

Discussion
Finding a suitable geographic location, where a species can germinate and be established successfully, required studying germination patterns under the influence of various abiotic factors. In this aspect mathematical model (TT, HT and HTT model) aids to determine and quantify the effect of these abiotic factors on seed germination time courses in seed lot [30]. Among these abiotic factors, T is one of the devastating stress affecting seed germinations in several plants [31,32]. Stress related to water (ѱ) is also major environmental factor limiting early seedling and seed germination [33,34]. Our results also demonstrated that seed germination was significantly affected by T and water stress (ѱ).
Our research findings showed that in Cicer arietinum G% significantly decreased at a temperature above and below T o (28˚C) (Fig 2C). The decreased in G% is due to the thermal denaturation of essential amino acids required for seed germination [35]. Similarly, the decreasing pattern of G% continued with decreasing water potential. The highest G% was recorded in -0.2 MPa as compared to control and lowest in -0.8 MPa at all the six constant temperatures (Fig 1A-1F). Decreasing ѱ resulted in the consumption of water potential energy and as a result, the supply of water is limited to seed [36]. Our germination findings against ѱ can be validated with the studies of [16] and [19] for watermelon and zucchini. The GR (g) showed a linear decreasing correlation with T up to T s �T o at constant ѱ, which is in agreement with the TT data on Hordeum sponataneum [37]. The T b obtained (4.7˚C) for chickpea in our experiment is lower than the given T b (7˚C) ( Table 3). The T b is an essential cardinal temperature required in developing a crop simulation model and suitable growing time [38].
The T o calculated in our study was 23˚C which is in agreement with [16]. Generally, seeds are incubated at a constant T in the laboratory, but they practice an inclusive range of T variations during the emergence time in the field. Seeds under changing T accumulated less thermal time than under constant T conditions [39]. The T c estimated (44.2˚C) in our study was probably equal to [40]. Based on the model behavior the cardinal temperature values were unaffected by ѱ in our experiment, which inversed with Mesgaran (Daucus carota) [39], (Retama raetam) https://doi.org/10.1371/journal.pone.0260990.g003 [41] and corroborated with Bakhshandeh [8]. The amount θH resulted from water potentials and germination fractions increased with increasing T at sub-optimal T. The GR (g) values plotted against T also showed a slight decline with increasing T at constant ѱ ( Table 1). The θH and GR (g) results are in contrast with the findings of [16] in Sesamum indicum and [31] in Allium tenuissimum. However, with reducing water potential the GR (g) values increased results of our experiment are in line with the findings of [42] for lemon balm and [20] for watermelon. In research studies and breeding programs, the θH values were used to categorize cultivars and varieties in terms of their sensitivity to ѱ. The values of base water potential at 50 percentiles ѱ b(50) showed an irregular pattern and were minimum (-1.28 MPa) at 21˚C and maximum at 42˚C (Table 2). Due to thermo-inhibition [20] also reported that ѱ b(50) values were minimal at sup optimal T and become positive (increasing) at supra optimal T. Results indicated that σC b values followed a liner probit trend of decreasing with increasing T up to T o and then irregular pattern at supra optimal T. This can be associated to a reduction in the activity of enzymes and oxygen demand during SG time. The σC b is used as an indicator to represent uniform changes in seed germination in seed lot [43]. However, based on estimated hydro time parameters (θH, C b(50) and σC b ) we can easily forecast whole germination time courses at any T and C.
The TT and HT concepts are successfully applied in many types of researches to explain SG behaviors at T (sub and supra) and C. But the TT concept showed an error at suboptimal T s by not predicting the decline in GR when T s >T o [30]. As a result, the HTT model was developed by Bradford to overcome the problem [44]. Now the HTT concept is a robust method to study how the interaction of environmental factors (T and C) influence SG in seed lot [20 and 30]. The hydrothermal time constant (θHTT) estimated for chickpea was 34.56 (MPa˚Ch -1 ) ( Table 3). Reduction in the seed germination characteristics such as radicle, plumule length, G%, GRI, GI, CVG, GE, SVI-I and II and TGI against high T and low ѱ have also been reported. High T and low C cause these reductions due to thermo-inhibition of cellular mechanisms and chemical reactions in the seed [44 and 45]. The influence of these stresses is most prominent in crop agronomy because these attributes signify the component of highest concern to farmers, as well as consumers [46,47]. Based on the statistical analysis, cardinal temperatures, θHTT and germination findings the HTT give an insight into the interactive effect of T and C on seed germination populations.

Conclusions and novelty perspectives
The results of our findings indicated that G% and GR were significantly affected by T, C and their interactions with accelerated aging periods. The HTT model developed has a clear physiological meaning and characterizes the response of C. arietinum SG behavior at all T s and C s . The GR and its attributes notably reduced across all the high T (35 and 42˚C) and reduced C (-0.8 MPa). Based on the TT, HT and models the highest θT1 and θT2 values were observed in -0.2 MPa at 28˚C, lowest and highest θH values at -0.8 MPa at 7˚C and -0.2 MPa at 28˚C, for HTT model predicted θHTT average value is 34.56 (MPa˚Ch -1 ), Ѱ b(50) is -1.34 MPa and σC b value 2.65 MPa at k T 0.104 MPa˚Ch -1 and cardinal T s (T b = 4.7˚C, T o = 23˚C and T c = 44.2˚C. According to available literature, no specific seed germination patterns research has been reported from Pakistan to date for Cicer arietinum using the hydrothermal time model. This is the first detailed study on any mathematical model to describe germination patterns of chickpea. While the germination parameters were also described for the first time through our study for chickpea under water stress and temperature. Eventually, the HTT model provides a detailed quantitative description of C. arietinum SG to develop the physiological status of chickpea seed populations.